**1. Soil Water Movement (water/vapor flow)**

In STEMMUS (**S**imultaneous **T**ransfer of **E**nergy, **M**ass and **M**omentum in **U**nsaturated **S**oil) model, the extended version of Richards’ equation with modifications made by Milly (1982) was employed to consider the vertical interactive process between atmosphere and soil. The dry air in the soil is considered to be a single phase. Thus not only diffusion, but also advection and dispersion is included in the water vapor transport mechanism. As for the liquid transport, the mechanism remains the same, but with atmospheric pressure acting as one of driving force. Root water uptake process is considered as a source sink term. The governing equation of the liquid and vapor flow can be expressed as:

Where *ρ*_{L }and *ρ _{V}* are the density of liquid water and water vapor;

*θ*and

_{L }*θ*

*are the volumetric water content (liquid and vapor); z is the vertical space coordinate;*

_{V}*q*and

_{L}*q*are the soil water fluxes of liquid and water vapor (positive upwards), respectively; and

_{V}*S*is the sink term for the root water extraction.

The liquid water flux is separated into isothermal (pressure head and soil air pressure gradient driven) and thermal (temperature gradient driven).

Where* K _{Lh}* and

*D*are the isothermal and thermal hydraulic conductivities, respectively;

_{Ta}*h*is the pressure head;

*γ*the specific weight of water;

_{w}*P*the mixed pore-air pressure (including vapor and dry air pressure) and

_{g}*T*is the soil temperature.

The water vapor flow in the unsaturated soil includes three ways: firstly diffusive transfer, driven by a vapor pressure gradient; secondly advective transfer, as part of the bulk flows of air and thirdly dispersive transfer, due to longitudinal dispersivity.

Considering vapor density to be a function of matric potential and temperature, the vapor flux can be divided into isothermal and thermal components.

Where *D _{Vh}* is the isothermal vapor conductivity; and

*D*is the thermal vapor diffusion coefficient,

_{VT}*D*is the advective vapor transfer coefficient, given in (Zeng et al., 2011).

_{Va}**2. Soil Air Flow**

Dry air transport in unsaturated soil is driven by two main gradients, the dry air concentration or density gradient and the air pressure gradient. The first one diffuses dry air in soil pores, while the second one causes advective flux of dry air. At the same time, the dispersive transfer of dry air should also be considered. In addition, to maintain mechanical and chemical equilibrium, a certain amount of dry air will dissolve into liquid according to Henry’s law.

Where *ε* is the soil porosity; *ρ _{da}* (kg m

^{−3}) is the density of dry air;

*S*(=1-

_{a}*S*) is the degree of air saturation in the soil;

_{L}*S*(=

_{L}*θ*/

_{L}*ε*) is the degree of saturation in the soil;

*H*is the Henry’s constant;

_{c}*D*(m

_{e}^{2}s

^{-1}) is the molecular diffusivity of water vapor in soil;

*K*(m

_{g}^{2}) is the intrinsic air permeability;

*μ*( kg m

_{a}^{-2}s

^{-1}) is the air viscosity;

*D*(m

_{Vg}^{2}s

^{-1}) is the gas phase longitudinal dispersion coefficient.

**3. Soil Heat Flow**

In the vadose zone, the mechanisms for heat transport include conduction and convection. The conductive heat transfer contains contributions from liquids, solids and gas. Conduction is the main mechanism for heat transfer in soil and contributes to the energy conservation by solids, liquids and air. Advective heat in soil is conveyed by liquid flux, vapor flux, and dry air flux. On the other hand, heat storage in soil includes the bulk volumetric heat content, the latent heat of vaporization and a source term associated with the exothermic process of wetting of a porous medium (integral heat of wetting) (de Vries, 1958).

where *C _{s}*,

*C*,

_{L}*C*and

_{V}*C*(J kg

_{a}^{−1 °C−1}) are the specific heat capacities of solids, liquid, water vapor and dry air, respectively;

*ρ*(kg m

_{s}^{−3}) is the density of solids;

*θ*is the volumetric fraction of solids in the soil;

_{s}*T*(°C) is the arbitrary reference temperature;

_{r }*L*(J kg

_{0}^{−1}) is the latent heat of vaporization of water at temperature

*T*;

_{r}*W*(J kg

^{−1}) is the differential heat of wetting (the amount of heat released when a small amount of free water is added to the soil matrix); and

*λ*(W m

_{eff}^{−1 °C−1}) is the effective thermal conductivity of the soil.

**4.Root water uptake process**

In STEMMUS, two alternative methods were used to elucidate the root water uptake: macroscopic and microscopic models.

On given potential transpiration, relative root length distribution and soil water pressure head, the macroscopic root water uptake model firstly distribute the potential transpiration into a monolayer according to the relative root length distribution.

Where *b*(*x*) is the normalized water uptake distribution, which describes the vertical variation of the potential extraction term, *S _{p}*, over the root zone, described in Šimůnek et al. (2008).

*T*is the potential transpiration.

_{p}Then the potential root water uptake is reduced to actual root water uptake as a function of the soil water potential.

The root water uptake term described by (Feddes et al., 1978) isWhere *a*(*h*) (dimensionless) is the reduction coefficient related to soil water potential.

The microscopic model, contrary to the macroscopic model, takes the gradient of pressure head from bulk soil to root surface and root radial conductance into account.

This root water uptake routine starts with the water extraction from a single root. The flow of water into a single root is described in two ways (P. de Willigen 2012): flow from bulk soil to root surface and flow from root surface into the root. The first flow is driven by the pressure head gradient from bulk soil to root surface.

Where Φ and Φ_{0} (cm^{-2} d^{-1}) the matric flux potential at bulk soil and root surface; *R*_{1,i }is the radius of root influence, which is a function of the root density (*R*_{1,i }= 1/π*L** _{rv,i}*),

*R*(cm) the root radius.

_{0}The other flow component, the flow of water from the root surface into the root, is proportional to the difference between the pressure head at the root surface and that in the root xylem.

Where *K _{R}* is the root conductance,

*h*and

_{0}*h*is the root pressure head at the root surface and in the xylem.

_{R}Assuming that there is no accumulation of water in the path of soil-root surface-xylem, it is obvious to conclude that the water flow from bulk soil to root surface equals the flow from root surface into the root.

* S _{i}*=

*U*, for

_{i}*i*=1,…

*n*

Following the principle of the continuous flow of soil-root surface-plant-atmosphere, root surface water pressure head can be related to the leaf water pressure head.

In the end, the summation of water flow into a single root equals actual transpiration.

Since the nonlinearity of these equations, iteration method is employed to solve for the unknown variables (*h _{0}*, to which Φ

_{0}relates,

*h*and

_{R}*h*) for a given bulk pressure head

_{L}*h*in the soil layer (to which Φ relates).

The detailed description of the basic soil physical processes can be found here.